What is the Wedge Product and How is it Defined?

[jdstokes]

I'm confused about the wording of this problem,

They define the wedge product by $u \wedge v = u\otimes v - v\otimes u$ but as far as I can tell this operation is not associative!

 

[George Jones]

I'm confused about the wording of this problem,

They define the wedge product by $u \wedge v = u\otimes v - v\otimes u$ but as far as I can tell this operation is not associative!

They have not tried to define an associative algebra, i.e., the exterior algebra, in this question.

They have defined meanings for the the abstract symbols $u \wedge v$, $u \wedge v \wedge w$, and $u \wedge v \wedge w \wedge x$, where $u$, $v$, $w$, and $x$ are all arbitrary vectors.

Their definitions can be extended to the definition of the exterior algebra, but this is not needed to do the question.

 

[jdstokes]

If the operation is not associative then an expression such as $u \wedge v \wedge w$ is not defined because the order of operations is not specified.

 

[George Jones]

If the operation is not associative then an expression such as $u \wedge v \wedge w$ is not defined because the order of operations is not specified.

They have defined what this symbol means. Don't think of it as the product of two wedges, think of it as one symbol.

Consider all possible permutations of the juxtaposed symbols $uvw$. Stick a + in front of the permutation if it is an even permutation of $uvw$ and and a - in front if the permutation is odd. Add all the terms, and insert tensor product symbols.